\(\int e^{c+b^2 x^2} x \text {erf}(b x) \, dx\) [66]

Optimal result

Integrand size = 17, antiderivative size = 37 \[ \int e^{c+b^2 x^2} x \text {erf}(b x) \, dx=-\frac {e^c x}{b \sqrt {\pi }}+\frac {e^{c+b^2 x^2} \text {erf}(b x)}{2 b^2} \]

[Out]

1/2*exp(b^2*x^2+c)*erf(b*x)/b^2-exp(c)*x/b/Pi^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {6517, 8} \[ \int e^{c+b^2 x^2} x \text {erf}(b x) \, dx=\frac {e^{b^2 x^2+c} \text {erf}(b x)}{2 b^2}-\frac {e^c x}{\sqrt {\pi } b} \]

[In]

Int[E^(c + b^2*x^2)*x*Erf[b*x],x]

[Out]

-((E^c*x)/(b*Sqrt[Pi])) + (E^(c + b^2*x^2)*Erf[b*x])/(2*b^2)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 6517

Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(a_.) + (b_.)*(x_)]*(x_), x_Symbol] :> Simp[E^(c + d*x^2)*(Erf[a + b*x]/(2*d)
), x] - Dist[b/(d*Sqrt[Pi]), Int[E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2), x], x] /; FreeQ[{a, b, c, d}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {e^{c+b^2 x^2} \text {erf}(b x)}{2 b^2}-\frac {\int e^c \, dx}{b \sqrt {\pi }} \\ & = -\frac {e^c x}{b \sqrt {\pi }}+\frac {e^{c+b^2 x^2} \text {erf}(b x)}{2 b^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.92 \[ \int e^{c+b^2 x^2} x \text {erf}(b x) \, dx=\frac {e^c \left (-\frac {2 b x}{\sqrt {\pi }}+e^{b^2 x^2} \text {erf}(b x)\right )}{2 b^2} \]

[In]

Integrate[E^(c + b^2*x^2)*x*Erf[b*x],x]

[Out]

(E^c*((-2*b*x)/Sqrt[Pi] + E^(b^2*x^2)*Erf[b*x]))/(2*b^2)

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.38

method	result	size
default	\(\frac {-2 \,{\mathrm e}^{b^{2} x^{2}+c} x \,{\mathrm e}^{-b^{2} x^{2}} b +{\mathrm e}^{b^{2} x^{2}+c} \operatorname {erf}\left (b x \right ) \sqrt {\pi }}{2 b^{2} \sqrt {\pi }}\)	\(51\)
parallelrisch	\(\frac {-2 \,{\mathrm e}^{b^{2} x^{2}+c} x \,{\mathrm e}^{-b^{2} x^{2}} b +{\mathrm e}^{b^{2} x^{2}+c} \operatorname {erf}\left (b x \right ) \sqrt {\pi }}{2 b^{2} \sqrt {\pi }}\)	\(51\)

[In]

int(exp(b^2*x^2+c)*x*erf(b*x),x,method=_RETURNVERBOSE)

[Out]

1/2*(-2*exp(b^2*x^2+c)*x*exp(-b^2*x^2)*b+exp(b^2*x^2+c)*erf(b*x)*Pi^(1/2))/b^2/Pi^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.95 \[ \int e^{c+b^2 x^2} x \text {erf}(b x) \, dx=-\frac {2 \, \sqrt {\pi } b x e^{c} - \pi \operatorname {erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{2 \, \pi b^{2}} \]

[In]

integrate(exp(b^2*x^2+c)*x*erf(b*x),x, algorithm="fricas")

[Out]

-1/2*(2*sqrt(pi)*b*x*e^c - pi*erf(b*x)*e^(b^2*x^2 + c))/(pi*b^2)

Sympy [A] (verification not implemented)

Time = 1.19 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.92 \[ \int e^{c+b^2 x^2} x \text {erf}(b x) \, dx=\begin {cases} - \frac {x e^{c}}{\sqrt {\pi } b} + \frac {e^{c} e^{b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{2 b^{2}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

integrate(exp(b**2*x**2+c)*x*erf(b*x),x)

[Out]

Piecewise((-x*exp(c)/(sqrt(pi)*b) + exp(c)*exp(b**2*x**2)*erf(b*x)/(2*b**2), Ne(b, 0)), (0, True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.92 \[ \int e^{c+b^2 x^2} x \text {erf}(b x) \, dx=-\frac {2 \, b x e^{c} - \sqrt {\pi } \operatorname {erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{2 \, \sqrt {\pi } b^{2}} \]

[In]

integrate(exp(b^2*x^2+c)*x*erf(b*x),x, algorithm="maxima")

[Out]

-1/2*(2*b*x*e^c - sqrt(pi)*erf(b*x)*e^(b^2*x^2 + c))/(sqrt(pi)*b^2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.84 \[ \int e^{c+b^2 x^2} x \text {erf}(b x) \, dx=-\frac {x e^{c}}{\sqrt {\pi } b} + \frac {\operatorname {erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{2 \, b^{2}} \]

[In]

integrate(exp(b^2*x^2+c)*x*erf(b*x),x, algorithm="giac")

[Out]

-x*e^c/(sqrt(pi)*b) + 1/2*erf(b*x)*e^(b^2*x^2 + c)/b^2

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.84 \[ \int e^{c+b^2 x^2} x \text {erf}(b x) \, dx=\frac {{\mathrm {e}}^{b^2\,x^2}\,{\mathrm {e}}^c\,\mathrm {erf}\left (b\,x\right )}{2\,b^2}-\frac {x\,{\mathrm {e}}^c}{b\,\sqrt {\pi }} \]

[In]

int(x*exp(c + b^2*x^2)*erf(b*x),x)

[Out]

(exp(b^2*x^2)*exp(c)*erf(b*x))/(2*b^2) - (x*exp(c))/(b*pi^(1/2))